Calculus Examples

$e^{5 x} \frac{d y}{d x} = 7$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $e^{5 x} \frac{d y}{d x} = 7$ by $e^{5 x}$ and simplify.

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Step 1.1.1

Divide each term in $e^{5 x} \frac{d y}{d x} = 7$ by $e^{5 x}$ .

$\frac{e^{5 x} \frac{d y}{d x}}{e^{5 x}} = \frac{7}{e^{5 x}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $e^{5 x}$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{e^{5 x} \frac{d y}{d x}}{e^{5 x}} = \frac{7}{e^{5 x}}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{7}{e^{5 x}}$

Step 1.2

Rewrite the equation.

$d y = \frac{7}{e^{5 x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{7}{e^{5 x}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{7}{e^{5 x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$y + C_{1} = 7 \int \frac{1}{e^{5 x}} d x$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Negate the exponent of $e^{5 x}$ and move it out of the denominator.

$y + C_{1} = 7 \int 1 {(e^{5 x})}^{- 1} d x$

Step 2.3.2.2

Simplify.

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Step 2.3.2.2.1

Multiply the exponents in ${(e^{5 x})}^{- 1}$ .

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Step 2.3.2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y + C_{1} = 7 \int 1 e^{5 x \cdot - 1} d x$

Step 2.3.2.2.1.2

Multiply $- 1$ by $5$ .

$y + C_{1} = 7 \int 1 e^{- 5 x} d x$

Step 2.3.2.2.2

Multiply $e^{- 5 x}$ by $1$ .

$y + C_{1} = 7 \int e^{- 5 x} d x$

Step 2.3.3

Let $u = - 5 x$ . Then $d u = - 5 d x$ , so $- \frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.3.1

Let $u = - 5 x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.3.1.1

Differentiate $- 5 x$ .

$\frac{d}{d x} [- 5 x]$

Step 2.3.3.1.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$- 5 \frac{d}{d x} [x]$

Step 2.3.3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 5 \cdot 1$

Step 2.3.3.1.4

Multiply $- 5$ by $1$ .

$- 5$

Step 2.3.3.2

Rewrite the problem using $u$ and $d u$ .

$y + C_{1} = 7 \int e^{u} \frac{1}{- 5} d u$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Move the negative in front of the fraction.

$y + C_{1} = 7 \int e^{u} (- \frac{1}{5}) d u$

Step 2.3.4.2

Combine $e^{u}$ and $\frac{1}{5}$ .

$y + C_{1} = 7 \int - \frac{e^{u}}{5} d u$

Step 2.3.5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$y + C_{1} = 7 (- \int \frac{e^{u}}{5} d u)$

Step 2.3.6

Multiply $- 1$ by $7$ .

$y + C_{1} = - 7 \int \frac{e^{u}}{5} d u$

Step 2.3.7

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$y + C_{1} = - 7 (\frac{1}{5} \int e^{u} d u)$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Combine $\frac{1}{5}$ and $- 7$ .

$y + C_{1} = \frac{- 7}{5} \int e^{u} d u$

Step 2.3.8.2

Move the negative in front of the fraction.

$y + C_{1} = - \frac{7}{5} \int e^{u} d u$

Step 2.3.9

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$y + C_{1} = - \frac{7}{5} (e^{u} + C_{2})$

Step 2.3.10

Simplify.

$y + C_{1} = - \frac{7}{5} e^{u} + C_{2}$

Step 2.3.11

Replace all occurrences of $u$ with $- 5 x$ .

$y + C_{1} = - \frac{7}{5} e^{- 5 x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = - \frac{7}{5} e^{- 5 x} + K$